## New subclasses of the class of close-to-convex functions

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- by Pran Nath Chichra
- Proc. Amer. Math. Soc.
**62**(1977), 37-43 - DOI: https://doi.org/10.1090/S0002-9939-1977-0425097-1
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## Abstract:

In this paper we introduce new subclasses of the class of close-to-convex functions. We call a regular function $f(z)$ an alpha-close-to-convex function if $(f(z)fâ€™(z)/z) \ne 0$ for*z*in

*E*and if for some nonnegative real number $\alpha$ there exists a starlike function $\phi (z) = z + \cdots$ such that \[ \operatorname {Re} \;\left [ {(1 - \alpha )\frac {{zfâ€™(z)}}{{\phi (z)}} + \alpha \frac {{(zfâ€™(z))â€™}}{{\phi â€™(z)}}} \right ] > 0\] for

*z*in

*E*. We have proved that all alpha-close-to-convex functions are close-to-convex and have obtained a few coefficient inequalities for $\alpha$-close-to-convex functions and an integral formula for constructing these functions. Let ${\mathfrak {F}_\alpha }$ be the class of regular and normalised functions $f(z)$ which satisfy $\operatorname {Re} \;(fâ€™(z) + \alpha zf''(z)) > 0$ for

*z*in

*E*. $f(z) \in {\mathfrak {F}_\alpha }$ gives $\operatorname {Re} fâ€™(z) > 0$ for

*z*in

*E*provided $\operatorname {Re} \alpha \geqslant 0$. A sharp radius of univalence of the class of functions $f(z)$ for which $zfâ€™(z) \in {\mathfrak {F}_\alpha }$ has also been obtained.

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## Bibliographic Information

- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**62**(1977), 37-43 - MSC: Primary 30A32
- DOI: https://doi.org/10.1090/S0002-9939-1977-0425097-1
- MathSciNet review: 0425097